2. Composite Hardness and Its Interpretation
Analyzing the information content of surface hardness has become important, as thin surface coatings (e.g., PVD, physical vapor deposition) have become more widely used. In these cases, determining the hardness of the surface film carries much uncertainty, as the hardness of the substrate also influences the measurement results. At this time, mathematical models, which interpreted surface hardness as so-called composite hardness, were developed. The measured composite hardness is a combination of substrate and surface film hardness. These models allow the estimation of the hardness surface film from the measured composite hardness if the hardness of the substrate is known.
Bückle [
1] was the first to propose a method to calculate composite hardness using the hardness of the substrate and the film. Jönsson and Hogmark’s [
3] model works on similar principles. It uses simple surface geometrical approach to separate the contribution of the substrate and the coating to the measured hardness, and the resultant hardness was derived from the relative projected sizes of the load-supporting areas under the indenter. Burnett and Rickerby [
4] published a novel method to determine composite hardness, which takes into account the elastoplastic behavior of the material under the indenter. They applied a volumetric mechanical model of indentation, which is based on the analogy with the expansion of a spherical cavity due to internal pressure. The theoretical background of this approach and its adaptation to indentation hardness testing have been known since the 1970s [
5,
6,
7]. Burnett et al. considered the volumetric ratio of film and substrate in the plastic zone in the calculation of composite hardness; therefore, this approach takes into account the plastic strain energy ratio of the coating and the matrix in the indentation process. Iost compared the Jönsson and Hogmark model and its improved version (Korsunsky model) [
8,
9] with the modified Puchi-Cabrera [
10] procedure and applied them to determine the actual hardness of thin layers [
11]. Coorevits and Mejias developed a method to determine in-depth hardness distribution, also based on the work of Jönnson and Hogmark [
12].
Khalay [
13] developed an artificial neural network-based model (ANNs) to predict the layer thickness of pre-nitrided steels. In another work, Khalaj and Pouraliakbara [
14] modelled layer thickness of treated coating by gene expression programming (GEP). A lot of research has dealt with the relationship between the plastic zone size and plastic depth during indentation. The fundamental expression of the relationship between the radius of the plastic deformation zone and the residual indentation depth was written by Lawn et al. [
15]. Giannakopoulos and Suresh [
16,
17] found that the radius of the plastic deformation zone can be expressed in terms of applied load and yield strength. Nayebi et al. [
18,
19] analyzed heat-treated tool steels with a gradient in hardness (yield strength), utilizing instrumented indentation and finite element models in order to predict the hardness versus depth profile of a heat-treated steel. Chen and Bull [
20] studied the relationship between the plastic zone radius and residual indentation depth, which was examined by using finite element analysis. Klecka et al. [
21] showed a rapid novel method to predict the gradients produced during surface heat treatment, which consisted of only a series of indentations at increasing loads on the sample surface requiring only a hardness tester and minimal sample preparation.
Detailed knowledge of this topic is also important since thin coatings have a great effect on the material’s performance or lifetime. There is a correlation between the microstructure and the mechanical properties (hardness and elastic modulus), impact resistance, and tribological performance of different systems.
For example, mono- and multi-layers have a very big effect on the lifetime and cutting ability of cutting tools [
22,
23,
24,
25]. The carbonized and nitrided surfaces also have a great effect to the microstructural, hardness, tribological, and corrosion behavior of parts [
26,
27,
28,
29].
In this paper, following Burnett’s approach, composite hardness is interpreted as a volumetric characteristic feature. The calculation of composite hardness is based on the volume of the hemispherical plastic zone, as shown in
Figure 2 [
4]. The applicability of this approach in the case of a Vickers-type indenter is also proved by Mata’s finite element computational results [
30].
According to the theory describing elastoplastic behavior during hardness testing (expansion of a spherical cavity caused by internal pressure), the relationship between radius
b of the hemisphere, which is the boundary surface of elastic–plastic deformation zones, and the half-diagonal
a of the Vickers indentation (
a =
d/2) is
where
E is the elastic modulus (GPa),
Hs is surface hardness (GPa),
p is a constant between 2 and 3 (in this study,
p = 3), and
φ is the indenter semi-angle (148°/2 = 74°). Equation (1) is of paramount importance in practice because it describes the connection between indentation size and the radius of the plastic zone. Due to the geometrical connection between diagonal
d and depth
h of the indentation,
b, the radius of the plastic zone can be interpreted directly as a function of
h indentation depth (
b =
b(
h)).
Burnett’s model was improved by Ichimura and applied to evaluate PVD layers of duplex-coated tool steels [
31,
32]. In order to take into account the effect of non-homogenous hardness distribution in the substrate, he divided the volume of the radius
b hemisphere of the plastic zone into
n + 1 spherical segments along planes parallel to the surface. Ichimura then calculated composite hardness in the following way:
where
Vf and
Hf are the volume and hardness of the surface film, respectively;
Vi and
Hi are the volume and characteristic hardness of the
ith spherical segment under the surface, respectively; and
V0 is the volume of the plastic zone of radius
b. If the surface film is neglected, then Equation (2) can be written as Equation (3) based on the generalization shown in
Figure 3. If the in-depth hardness distribution is appropriately known, the surface hardness belonging to an
h indentation depth can be estimated with Equations (1) and (3).
In Equation (3), i = 1 is the spherical segment in contact with the surface.
4. Materials
The relationship between surface hardness and the hardness distribution was studied on sections perpendicular to the surface of nitrocarburized hot working tool steel samples. The material of the samples was 1.2344 ESR/ESU electroslag remelted, high-purity hot working tool steel (C 0.39%, Cr 5.2%, Mo 1.4%, V 0.95%, Si 1.1%), with fine carbide distribution. This tool steel grade is used for the production of die casting tools, filling chambers, and injection molds for plastic.
The tool steel samples were heat treated according to the specifications of this quality as follows: austenitization in vacuum at 1050 °C; cooling in high-pressure nitrogen; and annealing in a nitrogen atmosphere three times at 530 °C, 600 °C, and 570 °C. After heat treatment, the average hardness of the samples was 50.6 HRc (521 HV). The specimens were nitrocarburized in a cyanide-free salt bath. After preheating at 380 °C for 60 min, the samples were nitrocarburized at 580 °C, for 60, 120, and 480 min for significantly different case depths. The target nitrided case depth was 0.08 mm (60 min), 0.1 mm (120 min), and 0.18 mm (480 min).
After heat treatment, the compound layer was removed chemically; accordingly, the continuous monotonic decreasing nature of the in-depth hardness function can be assumed.
In-depth hardness was measured on polished sections perpendicular to the surface, with a HV0.2 (1.962 N) load, with the Vickers method in steps of 50 μm. Surface hardness was also measured with the Vickers method, with increasing loads. For microhardness tests, loads of 0.2, 0.5, 1, and 2 kgf for macrohardness tests loads of 5, 10, 20, 30, 40, 60, 100, and 120 kgf were used. Hardness was tested with a Zwick 3212 (ZwickRoell GmbH & Co. KG, Ulm, Germany) and a conventional hardness tester. Three series of parallel measurements were performed.